The Ragnarok Riddle

Ragnarok: The fabled end of the world, when giants, monsters, and Norse gods battle for the future. The gods were winning until the great serpent Jörmungandr emerged. It swallowed Valhalla and contorted itself across the land. Odin has just enough power to strike with one final bolt of lightning, and you have the fabled hammer, Mjölnir. Can you two destroy the serpent? Dan Finkel shows how.

Transcript:

Ragnarok. The fabled end of the world, when giants, monsters, and Norse gods battle for the future.

The gods were winning handily until the great serpent Jörmungandr emerged. It swallowed Valhalla, contorted itself across the land, and then merged into one continuous body with no head and no tail. As it begins to digest Valhalla, an exhausted Odin explains that he has just enough power to strike the creature with one final bolt of lightning. If you magnify his blast with your fabled hammer, Mjölnir, it should pierce the massive serpent.

You’ll run with super-speed along the serpent’s body. When you hold your hammer high, Odin will strike it with lightning and split Jörmungandr open at that point. Then, you’ll need to continue running along its body until every part of it is destroyed. You can’t run over the same section twice or you’ll fall into the already blasted part of the snake. But you can make multiple passes through points where the creature intersects its own body. If you leave any portion un-zapped, Jörmungandr will magically regenerate, Odin’s last power will be spent, and Valhalla will fall forever.

What path can you take to destroy the serpent?

One powerful way to solve problems is to simplify. And in this case, we can focus our attention on the two things that are important for our path: intersections and the stretches of snake between them. Or, as they’re referred to in graph theory, nodes and edges. The edges are important because they’re what we need to travel. And the nodes matter because they connect the edges, and are where we may need to make choices as we run from edge to edge. This simplification into nodes and edges leaves us with a ubiquitous and important mathematical object known as a graph, or network. We just need to figure out how to travel what mathematicians call an Eulerian path, which traces every edge exactly once.

Instead of looking at the path as a whole, let’s zoom in on a single node. During some moment in your run, you’ll enter that node, and then exit it. That takes care of two edges. If you enter again, you’ll need to exit again too, which requires another pair of edges. So every point along your path will have edges that come in pairs. One edge in each pair will function as entrance; the other as exit. And that means that the number of edges coming out of every node must be even.

There are just two exceptions: the start and end points, where you can exit without entering, or vice versa. If we look at the network formed by the serpent again, and number how many edges emerge from each node, a pattern jumps out that fits what we just saw. Every node has an even number of edges emerging from it, except two. So one of these must be the start of your route, and the other the end.

Interestingly enough, any connected network that has exactly 2 nodes with an odd number of edges will also contain an Eulerian path. The same is true if there are no nodes with an odd number of edges— in that case the path starts and ends in the same spot.

So knowing that, let’s return to our full graph. We can begin by taking care of this edge here. Now we can zig-zag back and forth across the whole snake until we reach the end. And that’s just one solution— it helps to be systematic, but you’re likely to happen upon many others once you know where to begin and end your run.

You hold your hammer high at the opportune moment, and Odin sends the world-saving surge of lightning at you. Then you run like you’ve never run before. If you can pull this off, surely nothing could stop the might of the Norse Gods. And if something like that were out there, slouching its way towards you… well, that would be a story for another day.

Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.